Absorbing boundary conditions for inertial processos

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Absorbing boundary conditions for inertial random processes

Jaume Masoliver and Josep M. Porra`

Departament de Fı´sica Fonamental, Universitat de Barcelona, Diagonal, 647, 08028-Barcelona, Spain

Katja Lindenberg

Department of Chemistry and Biochemistry and Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0340

~Received 1 February 1996!

A recent paper by J. Heinrichs@Phys. Rev. E 48, 2397 ~1993!# presents analytic expressions for the first-passage times and the survival probability for a particle moving in a field of random correlated forces. We believe that the analysis there is flawed due to an improper use of boundary conditions. We compare that result, in the white noise limit, with the known exact expression of the mean exit time.@S1063-651X~96!06512-9# PACS number~s!: 02.50.2r, 05.40.1j

In two recent works@1,2# Heinrichs has studied some sta-tistical aspects of the motion of a free particle under the influence of a random acceleration modeled as Gaussian noise:

X¨ ~t!5F~t!, ~1!

where F(t) is Ornstein-Uhlenbeck noise, that is, a zero-mean Gaussian process with correlation function

^

F~t!F~t8!&5 D

2tc

e2ut2t8u/tc_, ~2!

where D is the noise intensity andtc is the correlation time.

The main results achieved in Ref. @1# have been to obtain approximate expressions for the joint probability density function, p(x,v,t), of the position x and the velocity v, and

for the marginal densities p(x,t) and p(v,t). These

approxi-mate results, valid for Ornstein-Uhlenbeck driving noise, can be generalized to obtain the exact expressions of p(x,v,t), p(x,t), and p(v,t) for Gaussian driving noises F(t) with

arbitrary correlation functions:

^

F~t!F~t8!&5k~t,t

8!.

~3!

Such a generalization has been carried out in Ref.@3#. Heinrichs’s second paper on the subject@2# used that ap-proximate expression for the marginal density of the position

p(x,t) to deal with the important and difficult problem of

obtaining the mean exit time~MET! T(x0) out of an interval

(0,L), for the ~initial! position x0 of the random particle, regardless of the value of the initial velocity @4#. We first observe that this approach is, at least, incomplete since the MET T(x0,v0) of an inertial process depends, in a

compli-cated manner, on both the position and the velocity of the

particle. Once we know the complete MET T(x0,v0), the

MET T(x₀) is given by the following average over all initial velocities @5,6#:

T~x0!5

E

2` `

p~v0!T~x0,v0!dv0, ~4!

where p(v0) is the distribution of initial velocities.

Hein-richs’s derivation of T(x₀) does not involve p(v0) and, in

fact, assumes that v050. As a consequence, what is

evalu-ated in@2# is the quantity T(x0,v050)[T(x0,0).

Moreover, we believe that the evaluation of the mean exit time at zero velocity, T(x0,0), is flawed due to the use of

improper boundary conditions for the problem. Indeed, in Heinrichs’s analysis the mean exit time T(x0) out of the

interval (0,L) is given by T~x0!5

E

0 ` dt

E

0 L p~x,tux0!dx, ~5!

where p(x,tux₀) obeys the Fokker-Planck equation ]p ]t5 1 2Dx~t! ]2_p ]x2, ~6!

with initial condition

p~x,0ux0!5d~x2x0!, ~7!

and absorbing boundary conditions

p~0,tux0!5p~L,tux0!50. ~8!

Equation ~6! is the Fokker-Planck equation for the mar-ginal density of the position ifv050. When v0Þ0 the

equa-tion for p(x,vux0,v0) reads@3#

PHYSICAL REVIEW E VOLUME 54, NUMBER 6 DECEMBER 1996

54

]p ]t52v0 ]p ]x1 1 2Dx~t! ]2_p ]x2, ~9!

and the marginal density of the position clearly depends on both the initial position x₀@cf. Eq. ~7!# and the initial veloc-ity v0. For a general Gaussian driving noise the function

D_x(t) is exactly given by@3# Dx~t!52

E

0 t dt

8

E

0 t8 dt

9~2t2t82t9

!k~t8,t

9!.

~10!

In the case of Ornstein-Uhlenbeck noise this function reads

@cf. Eq. ~2!#

Dx~t!5Dt@t2tc~12e2t/tc!#. ~11!

Finally, for Gaussian white noise tc[0 and

D_x~t!5Dt2_{. ~12!}

We now show that the boundary conditions~8! are not the appropriate absorbing boundary conditions for an inertial process such as ~1!. In effect, Eq. ~1! represents a two di-mensional random process (X,X˙ ), and the proper absorbing boundary conditions for trapping are @7–9#

p~L,v,t!50 if v<0, ~13!

p~0,v,t!50 if v>0. ~14!

These are the correct absorbing boundary conditions for the exit problem because the particle has never left the interval and it is impossible to find the particle with positive velocity at x50 or with negative velocity at x5L @10#.

Since the marginal density of the position is given by

p~x,t!5

E

2` ` p~x,v,t!dv, we see that p~L,t!5

E

0 ` p~L,v,t!dv.

Note that one does not know whether p(L,v,t) is zero or not

for all positive values of the velocity. Hence we may not conclude that p(L,t)50. Analogously

p~0,t!5

E

2`

0

p~0,v,t!dv,

and the assumption p(0,t)50 is also doubtful. We believe that Eq.~8! does not embody the correct absorbing boundary conditions for the random motion of an inertial particle. In fact, Eq. ~8! represents absorbing boundary conditions of some first-order diffusion process with a time dependent dif-fusion coefficient given by Eq.~10!.

One might argue that nonetheless these boundary condi-tions may lead to a good approximation for the mean exit time T(x0,0). We can check the goodness of such an

ap-proximation by comparing it with the exact result that we have for the case of Gaussian white noise. In effect, when

F(t) is Gaussian and d correlated noise we have recently shown that the exact expression for the mean exit time at zero velocity out of an interval (0,L) is @5,6#

T~x0,0!5N

S

2L 2 D

D

1/3

S

x0 L

D

1/6

S

12x0 L

D

1/6

F

F

S

1,21 3; 7 6; x0 L

D

1F

S

1,21 3; 7 6;12 x0 L

DG

, ~15!

where N5(4/3)1/6_{/G(7/3), and F(a,b;c;z) is the Gauss}

hy-pergeometric function@11#.

Let us now obtain an approximate expression for T(x,0) using the method of Ref. @2#. In the case of white driving noise the Fokker-Planck equation~6! reads

]p ]t5 1 2Dt 2] 2_p ]x2. ~16!

We will now solve this equation with the initial condition~7! and boundary conditions~8!. Let us write the solution to Eq.

~16! in the form of the following Fourier series:

p~x,tux0,0!5

(

n51

`

a_n~t!sin

S

npx

L

D

, ~17!

that immediately satisfies the boundary conditions ~8!. Sub-stituting Eq. ~17! into Eq. ~16! we see that the solution to

~16! with initial condition ~7! is

p~x,tux0,0!52 L n

(

51 ` e2nn2p2Dt3/6L2sin

S

npx0 L

D

sin

S

npx L

D

, ~18!

and the mean exit time reads@cf. Eq. ~5!#

FIG. 1. T(x0,0) as a function of x0, for noise intensity D51 and interval length L51. The solid curve corresponds to the approxi-mation given by Eq. ~19!. The dashed curve represents the exact result~15!.

T~x0,v050!5M

S

L 2 D

D

1/3

(

n₅₀ ` sin@~2n11!px0/L# ~2n11!5/3 , ~19!

where M527/3G(1/3)/32/3p5/3. This expression agrees with that of @2# in the white noise limitt_c→0. We note that Eq.

~19! predicts the correct dependence on the length of the

interval, that is, T(x0,0);L2/3 as L→` @cf. Eq. ~15! and

Ref. @5# #. In Fig. 1 we plot the ~approximate! result ~19! along with the exact result given by Eq.~15!. We clearly see

from this figure that the expression ~19! is a poor approxi-mation to the actual mean exit time. We therefore conclude that boundary conditions~8! are not suitable to deal with the escape problem for inertial processes.

This work has been supported in part by Direccio´n Gen-eral de Investigacio´n Cientı´fica y Te´cnica under Contract No. PB93-0812, by Societat Catalana de Fı´sica~Institut d’Estudis Catalans!, by the U.S. Department of Energy through Grant No. DE-FG03-86ER13606, and by the Gaspar de Portola` Catalonian Studies Program at the University of California.

@1# J. Heinrichs, Phys. Rev. E 47, 3007 ~1993!. @2# J. Heinrichs, Phys. Rev. E 48, 2397 ~1993!.

@3# J. Masoliver and K-G. Wang, Phys. Rev. E 51, 2987 ~1995!. @4# In fact, Heinrichs evaluates T(x0) out of the interval

(2L,L). This case can be trivially reduced to that of (0,L). @5# J. Masoliver and J. M. Porra`, Phys. Rev. Lett. 75, 189 ~1995!. @6# J. Masoliver and J. M. Porra`, Phys. Rev. E 53, 2243 ~1996!. @7# M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323

~1945!.

@8# T. W. Marshall and E. J. Watson, J. Phys. A 18, 3531 ~1985!. @9# J. M. Porra`, J. Masoliver, and K. Lindenberg, Phys. Rev. E 50,

1985~1994!.

@10# P. S. Hagan, C. R. Doering, and C. D. Levermore, SIAM J. Appl. Math. 49, 1480~1989!.

@11# Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun~Dover, New York, 1965!.